Electric Dipoles: Properties and Potential Solutions

[Schrodu]

Suppose we have a gas of randomly oriented (and in random motion) electric dipoles. Obviously the dipoles do not behave as free particles. How do we describe it's properties? Can we define potential etc. ?
I am looking for a solution without the large volume approximation. Is it possible to get an expression for the mean kinetic energy etc. of a particle given its total energy?

 

[pardesi]

will get some answers here shreyas.
though i strictly believe this isseroius mixture of eveidence and thermodynamic arguments

 

[Astronuc]

Look at this abstract -

Electron and anion mobility in low density hydrogen cyanide gas. I. Dipole-bound electron ground states

Two dipole molecules could form dimers or, but depending on density, dipole gases behave much like non-dipole (non-polar) gases.

 

[quetzalcoatl9]

you are talking about a truly many-body problem, where you must (usually) include the induced dipole-induced dipole interactions. if they don't matter, then why bother?

there are ways of doing this numerically, but they are difficult. you can start with an analytic expression by considering a dipole, mu=dq, interacting with a point charge, you'll then get

$$\nabla_\alpha \nabla_\beta \frac{1}{r}$$

as the induced field contribution, which needs to be solved for numerically for anything but the simplest of systems. that's why most molecular simulation techniques ignore induced dipoles.

 

[Schrodu]

there are ways of doing this numerically, but they are difficult. you can start with an analytic expression by considering a dipole, mu=dq, interacting with a point charge, you'll then get

$$\nabla_\alpha \nabla_\beta \frac{1}{r}$$

.

Could you clarify that a bit? I am not used to the standard notations. In my original attempt, I calculated the potential energy of two interacting dipoles in terms of their spacing and orientation. I want to average this out in some way (integrating over the angle obviosly gives net potential energy 0)

Thanks for the help.

 

[Sojourner01]

If you're looking for a simple solution, the only viable one is to treat it like a semi-ideal gas. The dipoles have translational, rotational and vibrational degrees of freedom, each of which is 1/2 kT per molecule in this simple approximation.

 

[quetzalcoatl9]

Could you clarify that a bit? I am not used to the standard notations. In my original attempt, I calculated the potential energy of two interacting dipoles in terms of their spacing and orientation. I want to average this out in some way (integrating over the angle obviosly gives net potential energy 0)

Thanks for the help.

the potential energy of interaction between a dipole and a point charge (if you draw out the two charges separated) that is far away is $$\nabla_a (\frac{1}{r})$$

since $$E = -\nabla V$$ then the dipoles contribution to the induced field is the expression i gave.

do a scholar google search for "molecular polarization" if you are more interested. there are review articles out there that summarize the field. also, Jackson's E&M book may interest you.

 

[Schrodu]

I found in this paper(page no. 4)

dipole field $$E=\frac{\mu}{4\pi \epsilon_0 r^3}$$ ...
... $$\frac{mv_w^2(r)}{2}=kT-(\frac{\mu_r^2}{4\pi \epsilon_0})(\frac{1}{d^3}-\frac{1}{r^3})$$

Shouldn't there be a term to take care of direction, $$U=\frac{\mu_r^2\cos\alpha}{4\pi \epsilon_0}$$?, $$\alpha$$ is angle between field and the axis

 

[genneth]

You can also consider a mean field approximation -- that will allow you to calculate the potential energy due to the orientation of the dipoles. In fact, with that approximation, I guess you'd get a hybrid free-gas / Curie ferromagnet effect, with a phase change due to the orientation. As far as dipole-dipole forces beyond merely torque, perhaps use the Van de Waals approximation?